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The Quarterly Journal of Mechanics and Applied Mathematics 1964 17(3):253-269; doi:10.1093/qjmam/17.3.253
© 1964 by Oxford University Press
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UPPER AND LOWER BOUNDS IN PROBLEMS OF MELTING OR SOLIDIFYING SLABS{dagger}

BRUNO A. BOLEY

( Columbia University )

The problem studied is that of a slab, heated in an arbitrary manner on one face and insulated on the other, which melts (or solidifies), the material being allowed to remain stationary after change of phase. Variable material properties are taken into account. After preliminary general considerations, it is shown that the solution to the stated problem is unique. It is then proved that higher rates of melting and higher temperatures will result from certain combinations of the magnitude of the applied heat input and of a fictitious heat source travelling with the solid-liquid interface. From this result a method is developed for the construction of upper and lower bounds to the solution of the problem; an example is also presented. It is also shown that, under the same arbitrary heat input, the rate of melting in the present problem is always lower than that in the companion problem in which the material is instantaneously removed after change of phase.


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